Use the properties of logs to write the expression as one log:

8log(base 5)4th square root of x minus 4log(base 5) x

To write the given expression as one log, we can use the properties of logarithms.

First, let's focus on the term "8log(base 5)4th square root of x." According to the power rule of logarithms, we can bring the exponent outside as a coefficient. Additionally, since the square root is equivalent to raising the number to the power of 1/2, we can rewrite it as follows:

8log(base 5)4th square root of x = 8 * log(base 5)(x^(1/4))

Next, let's consider the term "4log(base 5) x." Similar to the previous step, we can apply the power rule and write it as:

4log(base 5) x = log(base 5)(x^4)

Now, we can combine both terms into one logarithmic expression:

8 * log(base 5)(x^(1/4)) - 4 * log(base 5) x = log(base 5)(x^(1/4))^8 - log(base 5) x^4

Using the quotient rule of logarithms, we can simplify further:

log(base 5)((x^(1/4))^8 / x^4)

To simplify the expression inside the logarithm, we can use the power rule for exponents:

log(base 5)(x^2 / x^4) = log(base 5)(1 / x^2)

Therefore, the final expression as one log is:

log(base 5)(1 / x^2)